Discussing the article: "Two-sample Kolmogorov-Smirnov test as an indicator of time series non-stationarity" - page 5

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Евгений Черныш #:

As for the FDI, it most likely has exactly the same distribution for the SB as the Hurst index, i.e. normal. With the help of Monte Carlo method everything can be calculated there, Peters did just that in his work "Fractal Analysis of Financial Markets". FDI is no different from any other statistic in the sense that it itself is a random variable, like sample mean or sample variance, so you can easily find out how this statistic behaves on the SB, on small samples, on large samples, etc.

Maybe, but Peters has R/S statistics of sorts (read long ago). The distribution is not necessarily the same - different estimates of the same value can have different distributions. Peters has a good mathematical basis, but I don't agree with his conclusions - imho, the significance of the deviation from the SB is not that big.

 
Евгений Черныш #:

The requirement of equal distribution is good for theorem proving, rigorous proofs and within the Mathematical Statistics department, but for real data this requirement is too strict. You must control the course of the experiment, make sure that the conditions under which the observation of the random variable does not change over time. It is clear that in the case of stock quotes we do not control anything. We simply observe how the invisible hand of the market pulls a certain number (price increment) out of the box, but we do not know whether at each moment of time the contents of this box are changed or not (and no one will ever know). This is the reality and you have to work with what you have.

In my opinion, comparing day to day is correct, because we have Asian, European and American sessions in each sample. If I were to compare the Asian session with the American one, it would be wrong. Well, of course, everyone decides for himself.

In practice, you just need to estimate (approximately, of course) how much the theorem conditions are violated. And we should compare these violations with the significance of our results. For example, Perters did not do it and his deviations of prices from the SB (imho, of course) can be explained, for example, by volatility fluctuations.

Imho, the effect of deterministic volatility fluctuations (variance) should be removed, as this often leads to removing thick tails of distributions, which helps a lot. There was some article on the topic, Stepanov or something.

 
Евгений Черныш #:

I can and you can too, at least for model data.

Is the autoregressive process equally distributed? Identical.

Is it independent? No.

Does the Smirnov criterion "see" that? Yes.

We can't. The Autoregressive process is identically distributed in the sense of unconditional distribution, which cannot be recovered from a single implementation. Take the GARCH implementation, for example. It is a stationary process (by construction), but your Smirnov will define it as non-stationary by a single realisation.
 
I guess it's time to end this discussion which is way beyond the scope of the article. We will continue after your next article)
 
Aleksey Nikolayev #:

Imho, a clear problem with logic. A tautology from which something else is deduced.

Not at all.

1) The Smirnov criterion statistics will converge to the Kolmogorov distribution if the data are independent

2) If the data are dependent, the Smirnov statistic will have a different distribution from the Kolmogorov distribution.

Thanks to this difference you can judge whether there are dependencies in the data or not. At least for stationary distributions this is the case.

Everything is logical




 
Aleksey Nikolayev #:
We can't. The Autoregressive process is identically distributed in the sense of an unconditional distribution, which cannot be recovered from a single implementation. Take the implementation of GARCH, for example. It is a stationary process (by construction), but your Smirnov will define it as non-stationary by one implementation.
PDF SD ARCH(1)


I took the ARCH(1) model for simplicity.

As a result, the Smirnov criterion says that this is an independent, stationary (homogeneous) process, which was required to prove.

As you can see, Smirnov does not care about heavy tails of ARCH/GARCH distributions.



 
What's the point of all this? In financial markets, series are non-stationary. That's been known for a long time. So? Why prove it again?
 
Dmytryi Nazarchuk #:
What's the point of all this? In financial markets, series are non-stationary. That's been known for a long time. So? Why prove it again?

Perhaps to get a tool that will tell us where and when they are non-stationary. It is not possible to determine it all by eye, we need some criterion, that is what we are talking about.

 
Евгений Черныш #:
Not at all.

1) Smirnov criterion statistics will converge to the Kolmogorov distribution in the case of data independence

2) If the data are dependent, the Smirnov statistic will have a distribution different from the Kolmogorov distribution.

Thanks to this difference you can judge whether there are dependencies in the data or not. At least for stationary distributions it is so.

It makes sense




I see, we are talking about different things. I mean the test on two days, when we just get one statistic value, while you have a set of numbers on many pairs of days (as in the article).
 
Евгений Черныш #:


Took the ARCH(1) model for simplicity.

As a result, the Smirnov criterion says that this is an independent, stationary (homogeneous) process, which was required to prove.

As you can see, Smirnov does not care about heavy tails of ARCH/GARCH distributions.



Ok, I'll double-check it sometime. It's just that there was once a discussion that GARCH is stationary, although the realisations look non-stationary (in terms of variance?). I think there was non-stationarity when checking one implementation by some test.

PS It is very good that matstat specialists appear on the forum. Be sure to write more articles.

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